The optimization problem that we consider is minimizing the problem as follow:

min x ∈ R 1 n ∑ i = 1 n ρ τ ( 〈 a i , x 〉 2 − y i ) + λ ‖ x ‖ 1 / 2 1 / 2 , (2)

where

ρ τ ( t ) = { τ t ,       t ≥ 0 ( 1 − τ ) t , otherwise ,

x , y i , a i , n have been described in (1), τ ∈ ( 0,1 ) , 〈 ⋅ , ⋅ 〉 stands for the inner product. λ > 0 is the regularized parameter, and ‖ x ‖ 1 / 2 1 / 2 = ∑ i = 1 p | x i | 1 / 2 .

Noting that lim ‖ x ‖ 2 → ∞ ‖ x ‖ 1 / 2 1 / 2 → ∞ and λ > 0 , we obtain that

lim ‖ x ‖ 2 → ∞ 1 n ∑ i = 1 n     ρ τ ( ( a i T x ) 2 − y i ) + λ ‖ x ‖ 1 / 2 1 / 2 → ∞

which together with the continuity of the object function yields that Problem (2) has a bounded solution. To design an appropriate iterated algorithm, we give the following two lemmas.

Lemma 1. (see [16] [17]) The global solution t * of following problem has analytic expression,

arg   min t ∈ ℝ ( t − α ) 2 + μ | t | 1 / 2

t * = { f μ , 1 / 2 ( α ) , | α | > 54 3 4 μ 2 / 3 0,     otherwise ,

where

α ∈ ℝ ,   f μ , 1 / 2 ( α ) = 2 3 α ( 1 + cos ( 2 π 3 − 2 3 φ μ ( α ) ) )

with

φ μ ( α ) = arccos ( μ 8 ( | α | 3 ) − 3 2 ) .

Lemma 2. The global solution h * of following problem has analytic expression,

h * = arg min h ∈ ℝ μ 2 ( h − α 1 ) 2 + ρ τ ( h 2 − α 2 ) ,

where α 1 , α 2 ∈ ℝ , μ > 0 is a constant. It can be derived

h * = l μ ( α 1 , α 2 ) ,

where

l μ ( α 1 , α 2 ) = { α 1 1 + 2 τ μ , if   α 2 ≤ 0   or   α 2 > 0   and   | α 1 | > α 2 ( 1 + 2 τ μ ) α 2 , if   α 2 > 0   and   0 ≤ α 1 ≤ α 2 ( 1 + 2 τ μ ) − α 2 , if   α 2 > 0   and   − α 2 ( 1 + 2 τ μ ) ≤ α 1 ≤ 0 ,

0 < μ ≤ 2 ( 1 − τ ) and

l μ ( α 1 , α 2 ) = { α 1 1 + 2 τ μ , if   α 2 ≤ 0   or   α 2 > 0   and   | α 1 | > α 2 ( 1 + 2 τ μ ) α 1 1 − 2 ( 1 − τ ) μ , if   α 2 > 0   and   | α 1 | < α 2 ( 1 − 2 ( 1 − τ ) μ ) α 2 , if   α 2 > 0   and   α 2 ( 1 − 2 ( 1 − τ ) μ ) ≤ α 1 ≤ α 2 ( 1 + 2 τ μ ) − α 2 , if   α 2 > 0   and   − α 2 ( 1 + 2 τ μ ) ≤ α 1 ≤ − α 2 ( 1 − 2 ( 1 − τ ) μ ) , (3)

as μ > 2 ( 1 − τ ) .

One can get this lemma in the similar way of [18]. So, we omit the details.

We now employ the alternating direction method of multipliers (ADMM) algorithm to solve Problem (2). By introducing a pair of new variables, we reformulate Problem (2) as

min x , q , z 1 n ∑ j = 1 n ρ τ ( z i 2 − y i ) + λ ‖ q ‖ 1 / 2 1 / 2 s .t .       ( A I ) x − ( z q ) = 0, (4)

where z ∈ ℝ n and q ∈ ℝ p . Then, the augmented Lagrangian function of the above problem is

L r ( x , q , z ; Λ 1 , Λ 2 ) = 1 n ∑ i = 1 n     ρ τ ( z i − y i ) + λ ‖ q ‖ 1 / 2 1 / 2 + 〈 Λ 1 , x − q 〉     + r 2 ‖ x − q ‖ 2 2 + 〈 Λ 2 , A x − z 〉 + r 2 ‖ A x − z ‖ 2 2 , (5)

where Λ 1 , Λ 2 are the Lagrange multipliers, r is a positive constant.

Based on the framework of ADMM, the j + 1 th iteration can be calculated as

x j + 1 = arg min x L r ( x , q j , z j ; Λ 1 j , Λ 2 j ) , q j + 1 = arg min q L r ( x j + 1 , q , z j ; Λ 1 j , Λ 2 j ) , z j + 1 = arg min z L r ( x j + 1 , q j + 1 , z ; Λ 1 j , Λ 2 j ) , Λ 1 j + 1 = Λ 1 j + r ( x j + 1 − q j + 1 ) , Λ 2 j + 1 = Λ 2 j + r ( A x j + 1 − z j + 1 ) . (6)

In the following, we discuss the solution to each sub-minimization problem with respect to (w.r.t.) x , q , z .

1) Sub-minimization problem with respect to x: This problem can be written as

min x ∈ ℝ n 〈 Λ 1 j , x − q j 〉 + r 2 ‖ x − q j ‖ 2 2 + 〈 Λ 2 j , A x − z j 〉 + r 2 ‖ A x − z j ‖ 2 2 , (7)

Notice that

〈 Λ 1 j , x − q j 〉 + r 2 ‖ x − q j ‖ 2 2 + 〈 Λ 2 , A x − z j 〉 + r 2 ‖ A x − z j ‖ 2 2 = 〈 Λ 2 j , A x 〉 + 〈 Λ 1 j , x 〉 + r 2 ‖ A x − z j ‖ 2 2 + r 2 ‖ x − q j ‖ 2 2 . (8)

By the first order optimal condition, we can obtain the optimal solution. After analysis and comparison, we can conclude that

( r I + r A T A ) x j + 1 = r q j − A T Λ 2 j + r A T z j − Λ 1 j

where I stands for the identity matrix.

2) Sub-minimization problem with respect to q: This problem can be written as

min q ∈ ℝ n λ ‖ q ‖ 1 / 2 1 / 2 + 〈 Λ 1 j , x j + 1 − q 〉 + r 2 ‖ q − x j + 1 ‖ 2 2 . (9)

By simple calculation, one can see that the above problem is equivalent to the following minimization,

min q ∈ ℝ p λ ‖ q ‖ 1 / 2 1 / 2 + r 2 ‖ q − x j + 1 − Λ 1 j 2 ‖ 2 2 . (10)

According to the Lemma 1, we get

q d j + 1 = { f λ , 1 / 2 ( u d j ) , | u d j | > 54 3 4 λ 2 / 3 0,     otherwise ,

where u d j = ( x j + 1 + Λ 1 j / 2 ) d is the d-th element of the vector x j + 1 + Λ 1 j / 2 , and d = 1,2, ⋯ , p .

3) Sub-minimization problem with respect to z: This problem can be written as

min z i ∈ ℝ , i = 1,2, ⋯ , n 1 n ∑ i = 1 n ρ τ ( z i 2 − y i ) + r 2 ‖ z − W j ‖ 2 2 . (11)

where W j = A x j + 1 + Λ 2 j / r . For each i = 1 , 2 , ⋯ , n , the following problem

min z i ∈ ℝ 1 n ρ τ ( z i 2 − y i ) + r 2 ( z i − W i j ) 2 (12)

can be solved by l n r ( 1, y i ( W i j ) 2 ) W i j based upon Lemma 2. To the ease of presentation, we introduce the notation L n r ( 1, y / | W | 2 ) ∈ ℝ n whose i-th element is defined as l n r ( 1, y i ( W i j ) 2 ) . Therefore, the problem (11) can be solved by

z j + 1 = L n r ( 1, y / | W j | 2 ) ⊙ W j ,

where ⊙ denote the Hadamard product, respectively.

According to the above analysis, the iterative scheme for solving (4) can be given in Algorithm 1.

As discussed above, one can see that each subproblem of the proposed algorithm is well defined. Therefore, we discuss its convergence. Consider q-sub-problem

Algorithm 1. L_1/2QR PR: ADMM method for solving (4).

min q ∈ ℝ n 1 2 ‖ q − u ‖ 2 2 + μ ‖ q ‖ 1 / 2 1 / 2 (16)

where μ > 0 . According to [19], the first-order stationary point definition of (16) is given as follow.

Definition 1. Let q ^ be a vector in ℝ p and Q = Diag ( q ^ ) where Diag ( q ^ ) denotes a p × p diagonal matrix whose diagonal is formed by the vector q ^ . The vector q ^ is a first-order stationary point of (16) if

Q ( q ^ − u ) + μ 2 | q ^ | 1 / 2 = 0.

Similar to [20], we introduce the Karush-Kuhn-Tucker (KKT) conditions of the Lagrangian L r ( x , q , z ; Λ 1 , Λ 2 ) in (2) as follows

{ ∂ x L r ( x ˜ , q ˜ , z ˜ , Λ ˜ 1 , Λ ˜ 2 ) = 0, r Q ˜ ( q ˜ − x ˜ + Λ ˜ 1 r ) + λ 2 | q ˜ | 1 / 2 = 0, ∂ z L r ( x ˜ , q ˜ , z ˜ , Λ ˜ 1 , Λ ˜ 2 ) = 0, ∂ Λ 1 L r ( x ˜ , q ˜ , z ˜ , Λ ˜ 1 , Λ ˜ 2 ) = 0, ∂ Λ 2 L r ( x ˜ , q ˜ , z ˜ , Λ ˜ 1 , Λ ˜ 2 ) = 0 (17)

where ( x ˜ , q ˜ , z ˜ , Λ ˜ 1 , Λ ˜ 2 ) is a saddle point, ∂ represents the partial derivative, q ˜ = Diag ( q ˜ ) , q ˜ is the first-order stationary point of q-subproblem (10).

Since the Lagrangian L r ( x , q , z ; Λ 1 , Λ 2 ) is nonconvex w.r.t. x , q , z . After analyzing the first-order optimality conditions for the variable z and the subproblem (12), the above KKT conditions corresponding to these three variables can be described as

A H Λ ˜ 2 + Λ ˜ 1 = 0,

Λ ˜ 2 | q ˜ | 2 − Q ˜ Λ ˜ 1 = 0,

( 2 n r s i g n ( | z ˜ m | 2 ) − y m ) z ˜ m − ( Λ ˜ 2 r + A x ˜ ) m ∋ 0, m = 1,2,3 , ⋯ , n ,

z ˜ = A x ˜ ,

x ˜ = q ˜ ,

where

s i g n ( x ¯ ) = { 1, x ¯ > 0, − 1, x ¯ < 0, 0, x ¯ < 0.

Similar to Theorem 2.4 in [18], we show that our proposed algorithm converges to a saddle point satisfying KKT conditions.

Theorem 3. Assume that the successive differences of the two multipliers { Λ 1 j − Λ 1 j − 1 , Λ 2 j − Λ 2 j − 1 } converge to zero and { x j } is bounded. Then there exists a subsequence of iterative sequence of Algorithm 1 converging to an accumulation point that satisfies the KKT conditions of the saddle point problem (5).

We take p = 128 in all experiments, generating the true signal as Gaussian random sparse vector. The measurements matrix A generates from a i ∼ i . i . d . N ( 0, I ) . Regularization parameter λ is given by a fixed 10⁻⁴. The penalty parameter r is chosen as 10⁻².

For nonconvex problems, ADMM can converge to different (and in particular, nonoptimal) points, depending on the initial values and the penalty parameter [21]. We take Wirtinger flow [22] with initial point q 0 , which obeys dist ( q 0 , x ) ≤ 1 8 ‖ x ‖ , where dist ( q , x ) = min c = ± 1 ‖ q − c x ‖ 2 . Hence the algorithm proposed converges from the neighborhood of the global minimizer.

Like many phase retrieval methods, we use spectral initial values to achieve better recovery. To evaluate the efficiency of our method, we compare the successful recovery rate with 5 kinds of algorithm listed in Table 1.

We compare the recovery success rates of L_1/2QR PR at different values of τ in the noise-free case and noisy case including dense bounded noise and Laplace noise.

1) When not disturbed by noise, let τ = 0.1 , 0.2 , 0.4 , 0.5 , 0.6 , 0.8 , 0.9 . We report the success rates on Table 2 where the recovery is considered successful if the relative error is less than 0.0001.

2) When in dense bounded noise, let τ = 0.1 , 0.2 , 0.4 , 0.5 , 0.6 , 0.8 , 0.9 . The entries of the dense bounded noise are generated independently from U ( 0 , η max ) , where η max / ‖ x ‖ 2 = 0.01 . We say that the recovery is successful if the relative error is less than 0.001. Since the recovery success rate is 0 as τ = 0.1 , we here report the other cases on Table 3.

3) When in Laplace noise, let τ = 0.4 , 0.5 , 0.6 , 0.8 , 0.9 and n / p = 4 , 5 , 6 . The entries of Laplace noise are generated from Laplace ( 0 , μ max / 2 ) , where μ max ‖ y ‖ 2 / n = 0.001 . We say that the recovery is successful if the relative error is less than 0.005 and report the success rate on Table 4.

From Tables 2-4, one can see that the recovery success rate of L_1/2QR PR with τ = 0.5 is usually better than when τ equals other values. For Laplace noise, L_1/2QR PR with τ = 0.5 and τ = 0.8 both perform well when n / p = 6 .

Next, let’s continue to compare recovery success rate with the other five algorithms under different measurement fractions (n/p). We continue to use the same parameters and recovery success criteria as above. And let τ = 0.5 . Figure 1 shows how these algorithms behave without noise. Through repeated experiments, we find that median-MRWF has a better recovery success rate when measurement fractions (n/p) = 2, 3 and no noise interference in real field. The recovery success rate of L_1/2QR PR is slightly lower than that of median-RWF and median-MRWF, and higher than that of other algorithms. Figure 1 also shows how these algorithms behave when disturbed by noise. When receiving interference from dense bounded noise, the recovery success rate of L_1/2QR PR is higher than that of the other five algorithms in the real field. When receiving interference from laplace noise, the recovery success rate of L_1/2QR PR is higher than that of the other five algorithms in the real field.

In the following chapters, we make τ = 0.5 . And we fix the sparsity to s = 8 and use the competing algorithms listed in L_1/2QR PR the L₁ norm loss function, and L₀L₁PR adds L₀ regularization. The median-RWF and the median-MRAF of the L₂ norm loss function are highly robust to outliers through heuristic truncation rules. We compare the relative errors relative to the iteration count T under different measurement fractions (n/p), x is the true signal, x ^(t) is the tth iteration point.

Remark 3.1. We take n = 2p, 3p, 4p, 5p, 6p in real case. In detail, n = 2p and n = 4p are approximate theoretical sample complexity. When lad-admm returns to stability, we actually use n = 6p.

In the noise-free case, Figure 2 shows in real case, when n = 2p, L_1/2QR PR and L_1/2LAD PR have good recovery performance. When n = 3p, 4p, 5p, 6p, L_1/2QR PR is slightly better than other 5 algorithms.

Now, we consider the existence of dense bounded noise. The entries of the dense bounded noise are generated independently from U ( 0 , η max ) , where η max / ‖ x ‖ 2 = 0.001 , 0.01 . It can be seen from Figure 3, L_1/2QR PR shows great robustness to dense bounded noise in real case, while LAD-ADMM shows poor performance. L_1/2QR PR and L_1/2LAD PR have similar performance when n = 4p

in real case. Median-RWF and median-MRAF have similar performance when n ≥ 4p in real case. Another reasonable observation, we find the relative reconstruction error has 10 times increase as η shrinks by a factor of 10 for all algorithms.

Finally, we consider the presence of Laplace noise, the entries of Laplace noise are generated from Laplace ( 0 , μ max / 2 ) , where μ max ‖ y ‖ 2 / n = 0.001 , 0.01 . As can be observed in Figure 4, surprisingly, L_1/2QR PR and L_1/2LAD PR are very robust to Laplace noise, especially in real case, no matter when n = 2p, 3p, 4p, 5p, 6p. However, other methods show poor performance, even when n = 6p, LAD-ADMM, L₀L₁PR, median-RWF, median-MRAF don’t have satisfactory recovery. Another logical observation, we find the relative reconstruction error has 10 times increase as μ max shrinks by a factor of 10 for all algorithms.

The simulation results show that the L_1/2QR PR algorithm with quantile loss function and L_1/2 regularization has two significant advantages over other methods. One is the ability to recover the signal with fewer measured values, and the other is the robustness to asymmetrically distributed noise (such as dense bounded noise and Laplacian noise).